1

Profile Coordinate Metrology Based on Maximum Conformance

to Tolerances

Faculty of Engineering and Applied Science University of Ontario Institute of Technology

Oshawa, Ontario, Canada

Dr. Ahmad Barariahmad.barari@uoit.ca

CMSC - Coordinate Metrology Systems Conference (CMSC 2011), Phoenix, Arizona, 25 – 28 July 2011

II. Substitute Geometry Estimation (SGE)

The Best Substituted Geometry

εCi

eiεTi

Desired Geometry in the Reference coordinate System

pi

I. Point Measurement Planning (PMP)

Substituted Geometry in the Reference coordinate System

pi : i th Measured point

ri

Tolerance envelope

III. Deviation Zone Evaluation (DZE)

Probability Density Function of Geometric Deviations

f(e)

e

Three Basic Computation Tasks in Coordinate Metrology

3

Integration Inspection System: Current Research and Final Goal

SGEPMP

DZE

c) Integrated computational tasks

SGEPMP

DZE

b) Recent research on integration of tasks

PMP SGE DZE

a) Sequential tasks in traditional coordinate metrology

SGEPMP

DZE

d) Integrated Inspection System

Design & Manufact. Data

G Nominal geometryG′ Actual geometryG″ Substitute geometryG* Optimum substitute geometry

pi'

G*

Fitting Process

G″

pi ″

Δ(pi ′,G

″)

ΔG′ (G″)

G

G'

pi

εi

XZ

Y

1000

))cos(cos())cos(sin()sin(

))cos(sin(

))cos(cos(

621212

52,32,223

41,31,223

ζζζζζζ

ζζTζTζζ

ζζTζTζζ

ζT

))sin()sin(sin())cos(cos(

))sin()sin(cos())cos(sin(

123132,2

123131,2

ζζζζζζT

ζζζζζζT

))cos()sin(sin())sin(cos(

))cos()sin(cos())sin(sin(

123132,3

123131,3

ζζζζζζT

ζζζζζζT

Six-DOF rigid body transformation

Geometric Deviations

Substitute Geometry

Objective Function

Geometric Deviation

The Best Substituted Geometry

ei

Desired Geometry in the Reference coordinate System

pi

pi*

pi*: Corresponding point

pi: i th measured point

ni: normal vector at the corresponding point

DG: Desired geometry

T: Transformation matrix

Deviation Zone Evaluation of a Single Geometric Feature

GζTG

GζTpiζ

i

maxminObj

( ) GζTG ** ×=

*

**

i

ii

n

nppe ii *

i Gp ⊂*

6

Tolerance Zone & Residual Deviations

GpnnpG

GpnnpG

l

u

l

u

t

t

Tolerance Zone Definition(ASME Y14.5)

Upper tolerance limit

Lower tolerance limit

Normal vector

Gu ”

Gl

”

G'

pi'

XZ

Y

G ”pi″(u* ,v*)

Δ(p i′, G

” )

Res

idua

l D

evia

tion

Pi(u* ,v*)

7

A Drawback of Common Fitting Methods

A Measured Point that Complies to the Tolerance Zone

0

0

l

u

Gp

GpZone Tolerance p ,Gp

,Δ

,Δif

p1'

p 1 ″ Gu″

Gl″G″

Δ(p5′,G″)

p2'

p3' p4'

p5'p2 ″

XZ

Y

Measured pointCorresponding point

p1'

p1 ″

Gu″

Gl″

G″

Δ(p4′,G″)

p2'

p3' p4'

p5'p2 ″

XZ

Y

Measured pointCorresponding pointCorresponding point in the previous tolerance zone

Accept

Reject

8

Application: Over-Cut & Under-Cut in Closed Loop Machining

G'

pi'

XZ

Y

G *pi″

Δ(p i′, G

* )

Over-Cut

Und

er-C

ut

Under-Cut

G

Closed-Loop Machining Strategy:

Correction of Under-Cut Regions Elimination of Over-Cut Regions

Common fitting methods are not suitable for closed-loop machining & inspection.

9

Required Properties for the Fitting Function

Fitting to the tolerance zone and not to the nominal geometryFitting to eliminate the over-cut situation Fitting to minimize the under-cut by minimizing the residual deviations

p/1n

i

p

ip en

1C

0,Δif

0,Δif,Δ

0,Δ0,Δif0

e

i

ii

ii

i

l

uu

lu

Gp

GpGp

GpGp

Residual Deviation Function

Minimization Objective

Fitting Function

G

G'pi'

pi ″

Δ(p i′,

G l″)

Δ G′(

G″)

G u

G l

G″Gu″

Gl″

Δ(pi′,Gu″)

XZ

Y

G

G'

pi'

XZ

Y

G u

G l

Gu *

Gl

*

G *

Λ G′(Gu

* )

pi″

Maximum Conformance to Tolerance (MCT) function (p→∞)

ii

emaxC

10

Objectives in Closed-Loop Inspection and Machining

Fitting criteria for Closed-Loop Machining & Inspection:

Inspection Based on Machining:

Machining Based on Inspection:

To develop a fitting methodology to construct the substitute geometry that minimizes the required compensation operations but maximizes the compensation capability of the geometric deviations.

To develop a method to determine number and location of the measured points based on the characteristics and properties of the actual machined surface, to reduce uncertainty of the process.

To develop a compensation procedure based on the inspection results. The procedure should be capable to interpolate the compensation requirements between the measured points for the entire machined surface.

11

Barrier ensures that a feasible solution never becomes infeasible. However, this objective function can by highly non-linear with discontinuities.

In practice, the optimization may have an infeasible initial condition and stuck there.It is likely to be stuck in a feasible pocket with a local minimum.

Modification of MCT Function

Two drawbacks of this objective function are:

Adding a penalty condition instead of the barrier condition to avoid straying too far from the feasible solutions. Utilizing a method for iterative data capturing to escape from the local

minima by increasing the energy level.

Solutions:

12

General Form of Penalty Function (Juliff, 1993; Patton et al., 1995; Back et al., 1997)

Penalty Function

xx MCP P

Distance metric

function

Penalty factor

satisfiedisintconstraif0

violatedisintconstraif1P

0

0

000

ll

uu

lu

GpGp

GpGp

GpGp

,Δif,ΔC

,Δif,Δ

,Δ,Δif

e

ii

ii

ii

i

For any point with the over cut condition the Δ(pi',Gl″) is a negative value that monotonically decreases when the point moves further from the tolerance zone. Therefore it can be a good choice for the distance metric function.

Modified Residual Deviation Function

Gradi

ent

Transformation of Substitute Geometry

Pena

lized

zone

Res

idu

al D

evia

tion

Fu

nct

ion

MCT Function

i

iemaxminCminObj

ζζ

13

Adaptive Penalty Function

Error Model-Local two directional sinusoid waves

14

Distribution of Geometric Deviations

Error Model-Local two directional sinusoid waves

MDZ MCTHalf- Normal Distribution

xx2)x(f

x2

x

dttx,e2

1x

2

Shape factor

α=0 : standard normal densityα→∞: half-normal density

1,01 2

15

Penalty factor, C, controls the velocity of transition from the state of the standard normal to the target state of extremely skewed

Violation of the Feasible Solution Area

Ct

Penalty Factor

Very Sever Penalty Function Convergence problem

C

,Δe

ii

max *uGp

Violation of the Feasible Solution

Large

Small

selecting a relatively small penalty factor (C=10).compensate the violation of the feasible solution by adoption of lower

tolerance limit

Solutions:

ett ll

16

Distribution of the Geometric Deviations (DGD)

Problem 2: is the region with maximum deviation sampled?

p'

Problem 1: how much is the deviation of an unmeasured point?

Approach: Search-Guided Sampling (Adoptive)Assumption: Distribution of deviations on the manufactured surface has a continues Probability Density Function .

Approach: Using Surface’s Geometric CharacteristicsAssumption: Gradient of the deviations is a direct function of the proximity of the Surface points with a high confidence level.

17

Example: Effect of Systematic Machining Errors

Kinematic Modeling of Generic Orthogonal Machine Tools Using Homogeneous Transformation Matrices

Homogeneous Transformation of X-AxisIMxAxHx

misalignment homogeneous

matrix

Motion homogeneous

matrix

Identity matrix

Homogeneous Transformation of Workpiece HzHy,Hx,Hw 1f

Homogeneous Transformation of Tool HzHy,Hx,Ht 2f Actual Machined Point

hwHwtHtp offsetoffset

Zero offset Homogenous vector

18

NURBS Presentation of Machined Surface

Quasistatic Linear Operator

Jacobian Matrix of the Actual Machining Point

1000

3GY

3

GY

3

GX

3

2GY

2

GY

2

GX

2

1GY

1

GY

1

GX

1

pppp

pppp

pppp

p

pJ

ThphIJΩ

Explicit Form of Machined Geometry

GΩG

NURBS Presentation of the Machined Surface:

1vu,0vu,vu,m

0i

n

0jji,ji,

QΩRG

NURBS piecewise rational basis

functions

Matrix of the surface control points

19

Behavior of Systematic ErrorsA Typical Vertical Machining Center

(Calibration using laser interferometer, electronic levels, optical squares)

00000.100000.000000.000000.0

0.02549-1.000000.0000200000.0

0.11600-0.000001.0000000000.0

0.05100-0.000000.00005-00000.1

XYWZTΩ

A Typical Horizontal Machining Center (Calibration using laser interferometer, electronic

levels, optical squares)

00000.100000.000000.000000.0

0.025001.000000.0000000000.0

0.024000.00006-1.0000000000.0

0.010000.000040.0000000000.1

XWYZTΩ

Nominal Geometry

20

Geometric Deviations Resulting from Systematic ErrorsGeometric Deviations

Hor

izon

tal M

achi

ning

Cen

ter

Systematic Error Vectors

Vert

ical

Mac

hini

ng C

ente

r

21

Search-Guided Sampling

Monitoring Continuity in Probability Density Function (PDF) of Geometric Deviations

eee d)(fRgPrRg

i

On-Line Estimation of PDFIn contrast, the only assumption in this works is the continuity of the true probability density function.

Probability Pr that a given deviation ei

will fall in a region Rg

Window Function 2

x2

e2

1x

Range of Windowing

hh

h

floori

i

)2min

(e

e1

h

h

h

ceili

i

)2max

(e

em

Parzen Windows Method (Parzen, 1962 )

n

1i

in hh

1

n

1f

eee

Window width

22

Iterative Search Hessian Function

Positive Maximum Absolute Hessian

mjh

)(f)(f2)(f

1m,...,2jh

)(f)(f)(f2)(f

1jh

)(f)(f2)(f

2

1mmm

2

1j1jjj

2

211

eee

eeee

eee

Negative Maximum Absolute Hessian

23

Fitting Uncertainty Using the Search Method Error Models (magnification: 100×):

1-Quasistatic errors of a vertical machine tool

2-One directional sinusoid wave

3-Two directional sinusoid waves 4-Local two directional sinusoid waves

24

Stratified Sampling

Error Model-Local two directional sinusoid waves

144 Stratified Points64 Stratified Points64 Random Points

25

Result of Search-Guided Sampling

Error Model-Local two directional sinusoid waves

26

Estimation of Uncertainty-Results

100 Times MiniMax Inspection Using Five Different Data Capturing Method (2000 Experiments)

27

Plug-In Uncertainty – Bootstrap Estimation

Plug-in uncertainty comes from the fact that it is always unknown how much of the captured dataset is a good representation of the real distribution function

The plug-in uncertainty is very much related to the probability of capturing critical points. A Bootstrap method is used to evaluate this probability.

Estimation of Maximum Geometric Deviation

P=(P1, P2, P3,…,Pn) and Θ =θ(P)Empirical probability density function of f: =1/n on each of the observed values

*n

*3

*2

*1

* P,...,P,P,PPf~

B,...,3,2,1kP~ k**

k

Bootstrap estimate of the standard error

21

B

1k

2*av

*k

*~

~~1B

1sd

B

~~

B

1k

*k

*av

28

Estimation of Plug-In Uncertainty-Bootstrap Results

100 Bootstrap Replications of Inspection of Five Different Data Capturing Method (2000 Experiments)

29

Distribution of the Geometric Deviations (DGD)

Pragmatic Space )v,u(GG

X

Z

Y

Cartesian Workspace Space

p'

Parametric Spaceu

e

v

Mapping

u*

v*

es

30

Interpolation of Geometric Deviations

Recall:

the variations of non-rigid transformation vectors of the machined point has a direct relationship with the distance of the nominal points.

A Proximity Problem

SxxsxsxsV jii ,ij:

Voronoi Diagram

Delaunay Triangle (O’Rourke, 1998; Okabe at el., 2000)

Variation of Non-Rigid Body Transformations

pζT-Ω

pζT-ΩppζT-Ωε*

**n

31

Interpolation Procedure

Position in the Parametric Space:

A Location between Sites any location on the uv parametric plane belongs to an individual Delaunay triangle

u

e

vek

sjsi

sk

oi oj

ok

ejei

er

r

0

-eevvuu

-eevvuu

-eevvuu

ik

ij

i

ikik

ijij

ii

ssss

ssss

ss

ir eC

BAe

uuvvvvuuC

vv-ee-eevvuuB

uu-ee-eeuuvvA

ijik

ijik

ikijikij

ikiji

ikiji

ssssssss

sssssr

sssssr

32

Case Study: DGD of a NURBS surface

Stamping Die of front door of a vehicle with the general dimensions of 1150mm×1080mm×35mm

(Forth order uniform, non-periodic NURBS surface with 16 control points)

Tolerance Specification

33

Simulation of Machining (Vertical Machine Tool)

PDF of Residual DeviationsInspection (Search procedure captures in 163 data points)

Step 1

Step 2

Mean of geometric deviation (mm)

Standard deviation of

geometricdeviation

Maximum geometric deviation Minimum geometric deviation

Deviation (mm)Parameters in the substitute surface

[u v]

Deviation (mm)

Parameters in the substitute surface

[u v]

0.031126 0.031716 0.083133 [0.0201 0.0211] -0.040000 [0.0229 0.9629]

34

Interpolation of Deviations

Development of DRD

Step 3

35

Inspection

Machining

Application: Closed-Loop of Machining and Inspection

Brown & Sharpe CMM Renishaw PH9 Probe HeadHorizontal Spindle Rotary TableReported displacement accuracies: ±0.004mmRoom Temperature: 20°C

Whali CNC Machine Tool Five AxesHorizontal Spindle Rotary TableReported Positional Accuracy: ±0.015mmRoom Temperature: 20°CMaterial:Aluminum 6061 Coolant: OilTool: 12mm Ball-noseDepth of Cut: 0.25mmft= 0.015(mm/min) and V=110 (m/min)

)rpm(3000)mm(12

min)/m(1101000)rpm(N s

min)/mm(90

)rpm(30002)tooth/mm(015.0min)/mm(fm

36

Setup & Machining Phase

Alignment Setup Roughing

Reference Patch Finishing CC-Lines Finished Part

37

Inspection Phase

Physical Measurement Cylindrical Fit DGD

Virtual Data Capturing Final Inspection

38

Experiment #1:Flexible Knot LocationsBefore

After

%8.73)mm(0332.0

)mm(0087.0)mm(0332.0

RangeErrorTotalofductionRe

39

Experiment #2- First Degree NURB Surface

Control Net Second Degree NURBS Surface

First Degree NURBS SurfaceCC-Lines

Upper Tolerance=0.006 mmLower Tolerance=-0.007 mm

40

Setup & Machining Phase

Alignment Setup Roughing

Reference Patch Finishing CC-Lines Finished Part

41

Inspection Phase

Physical Measurement Cylindrical Fit DGD

Virtual Data Capturing Final Inspection

42

Experiment #ResultsBefore

After%2.64

)mm(0358.0

)mm(0128.0)mm(0358.0

RangeErrorTotalofductionRe

43

Conclusions

A new fitting methodology for coordinate methodology is developed that maximizes conformance of the measured points to a given tolerance zone.

Generating detailed information of the deviation zone on the measured surfaces should be based on the needs of the upstream processes such as compensating machining, finishing or reverse engineering.

A methodology is developed to estimate distribution of the geometric deviations on a surface that is measured using discrete point sampling.

Developed search method is an alternative approach in coordinate data capturing which significantly reduces plug-in uncertainty.

Integration of computational tasks in coordinate metrology significantly reduces measurement uncertainties.